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Hydrodynamics of Cavitation
Published in Dmitry A. Biryukov, Denis N. Gerasimov, Eugeny I. Yutin, Cavitation and Associated Phenomena, 2021
Dmitry A. Biryukov, Denis N. Gerasimov, Eugeny I. Yutin
Complex velocity can be represented on a complex plane as a corresponding vector (originated from the origin, so to say); a hodograph is the line formed by the ends of this vector taken at different moments of time or at different spatial points which correspond to various values θ.
Van Deemter’s analysis of drainage to incised ditches in lowland areas
Published in Silke Wieprecht, Stefan Haun, Karolin Weber, Markus Noack, Kristina Terheiden, River Sedimentation, 2016
Briefly, the analysis is based on the flow in an infinitely deep, homogeneous, isotropic aquifer subjected to a constant, uniform rainfall intensity of N mm/day. N is a negative number in cases of rainfall. The hydraulic conductivity of the soil material is K mm/day and the flow region is symmetric to the adjoining flow regions with identical geometric characteristics. Therefore, the flow regime can be addressed with the potential flow theory methodology using conformal transformations. In this approach, the complex spatial plane, is represented by the expression z = x + iy where x and y are the spatial coordinates of the flow region. This spatial plane is projected onto the upper half of a complex plane, say t, in such a way that the vertices of the flow region are located on the real axis of this complex plane. Likewise, the complex flow of this field, given by the expression ω = ϕ + iψ, where ϕ and ψ are the complex pressure and streaming potentials, are projected on the upper half of a complex plane through a series of conformal transformation. Again, the complex potentials at the vertices are again located on the real axis of the t-plane. Then, a one-to-one projection of the conformal transformed spatial and flow potential field is sought through a linear fractional transformation of the transformed z-plane and ω-plane. The vertices of the flow field are the endpoints of line segments representing the boundaries (open or closed) of the flow field. In this way, the upper half of the complex t-plane represents the transformed flow region and the functional relationship ω(t) and z(t) from which ω = f(z) can in principle be obtained. Fixed boundaries of the flow field usually are streamlines, while at open boundaries the atmospheric pressure is set at zero and the pressure potential is then determined by the gravitational potential. At each point in the flow field the flow has a complex velocity W = u + iv. The graph consisting of the component velocities of the flow field commonly referred to as the Hodograph. Of particular interest and usefulness in the hodograph is the velocity representation of the open boundary.
Modified pseudo-dynamic analysis of slope considering logarithmic spiral failure surface with numerical solution
Published in Australian Journal of Civil Engineering, 2022
Suman Hazari, Sima Ghosh, Richi Prasad Sharma
Soil slope may be natural or artificial, which becomes unstable and sometimes the failure may be catastrophic. Different methods used in the slope stability analysis are the limit equilibrium method, limit analysis method, numerical modelling, etc. The limit equilibrium method satisfies force and/or moment equilibrium of a soil mass considering the potential failure wedges (Fellenius 1936; Bishop 1955; Spencer 1967). Limit analysis has been investigated due to its physical significance and strict solving range considering velocity hodograph (Chen, Giger, and Fang 1969; Michalowski 2010; Utili 2013). Numerical modelling has emerged as a tool based on the application of finite element and upper bound analysis of limit analysis (Ugai and Leshichinsky, 1995; Griffiths and Lane 1999).
Current meter methodology for discharge measurement in circular pipe
Published in ISH Journal of Hydraulic Engineering, 2023
Cristian Purece, Valeriu Panaitescu
In general, this discharge measurement method can be applied if a certain amount of symmetry of the velocity spatial hodograph is ensured. That is why ASME PTC 18 (2002) and IEC 41 (1991), measurement codes recommend ensuring minimum 20D pipeline length upstream and 5D downstream the measuring section S. The criterion for the verification of achieving this symmetry is the comparison by means of planimetry of the areas ω1-3 and ω2-4 of the spatial holographs of the velocities by the two diameters D1-3 and D2-4 current meter location support in the measuring section (Figure 4).
A fractionally magnetized flow of force fields and Fermi–Walker conformable derivative on the unit sphere
Published in Waves in Random and Complex Media, 2022
Talat Körpinar, Rıdvan Cem Demirkol, Zeliha Körpınar
In the most general case, the uniform acceleration of a point particle is described by the time rate of variation of its acceleration, when it is measured in a rest instantaneous mechanism, vanishes identically. This definition is generally translated into the covariant differential equations satisfying special relations. This system is also interpreted as the equation of the moving fictitious particle in velocity four-space; the hodograph of the motion, i.e. the trajectory of the given particle, being a geodesic of the non-ordinary velocity four-space.